Question 1

In the figure,? and ? are positive angles.?  ? a = 9,b = 18
Write ? as a function of ?.
?
A)? = arcsin(9sin?)
B)? = sin( $\frac { 1 } { 2 }$ sin?) ?
C)? = arcsin( $\frac { 1 } { 2 }$ sin?)
D)? = sin( $\frac { 1 } { 2 }$ sin?)
E)? = arcsin( $\frac { 1 } { 2 }$ sin?)

Accepted Answer

The answer of In the figure,? and ? are positive...

Question 2

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. A = 129°,b = 8,and c = 11
A)34.19 sq.units
B)41.03 sq.units
C)37.61 sq.units
D)30.77 sq.units
E)27.36 sq.units

Accepted Answer

The area of a triangle is given by the formula A = (1/2) * b * c * sin(A), where A is the angle opposite side a, b is the length of side b, and c is the length of side c. In this case, A = 129°, b = 8, and c = 11, so the area of the triangle is A = (1/2) * 8 * 11 * sin(129°) = 34.19 sq. units.

Question 3

In the figure,? and ? are positive angles.?  ? a = 3,b = 12
Use ? = arcsin(0.25sin?)to write c as a function of ?.
?
A) $c = \frac { 12 \sin [ \pi - \beta - \arcsin ( 0.25 \beta ) ] } { \sin \beta }$
B) $c = \frac { 12 \sin [ \pi + \beta - \arcsin ( 0.25 \beta ) ] } { \sin \beta }$
C) $c = \frac { 0.25 \sin [ \pi - \beta - \arcsin ( 12 \beta ) ] } { \sin \beta }$
D) $c = \frac { 12 \sin [ \pi + \beta + \arcsin ( 0.25 \beta ) ] } { \sin \beta }$
E) $c = \frac { 12 \sin [ \pi - \beta + \arcsin ( 0.25 \beta ) ] } { \sin \beta }$

Accepted Answer

The answer of In the figure,? and ? are positive...

Question 4

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. A = 118 ${}^\circ$ ,b = 9 and c = 8.
?
A)38.14 sq.units
B)31.79 sq.units
C)25.43 sq.units
D)28.61 sq.units
E)34.96 sq.units

Accepted Answer

The area of a triangle can be found using the formula $Area = \frac{1}{2}bc\sin(A)$, where $b$ and $c$ are the lengths of two sides, and $A$ is the angle between those sides. Plugging in the given values, $b = 9$, $c = 8$, and $A = 118^\circ$, we get $Area = \frac{1}{2} \times 9 \times 8 \times \sin(118^\circ)$. Calculating this gives an area of approximately 31.79 square units.

Question 5

A straight road makes an angle,A,of 10° with the horizontal.When the angle of elevation,B,of the sun is 56°,a vertical pole beside the road casts a shadow 8 feet long parallel to the road,see figure.Approximate the length of the pole.Round answer to two decimal places. 
A)2.48 feet
B)10.29 feet
C)6.22 feet
D)11.86 feet
E)6.94 feet

Accepted Answer

First, we need to find the height of the pole by using the shadow and the angle of elevation of the sun.

Let h be the height of the pole.

tan(56°) = h/d, where d is the distance between the top of the pole and the end of the shadow.

We know that the shadow is 8 feet long and parallel to the road, and the angle between the road and the horizontal is 10°, so:

d = 8/cos(10°)

Plugging this into the equation above, we get:

h = tan(56°) * 8/cos(10°) = 10.29 feet (approx.)

Therefore, the approximate length of the pole is 10.29 feet, which is closest to choice B.

Question 6

A park ranger at point A observes a fire in the direction N25°26'E.Another ranger at point B,4 miles due east of A,sites the same fire at N58°34'W.Determine the distance from point B to the fire.Round answer to two decimal places.
A)1.73 miles
B)3.63 miles
C)2.05 miles
D)4.08 miles
E)3.43 miles

Accepted Answer

The answer of A park ranger at point A observes...

Question 7

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. C = 55°43',a = 6.5,and b = 14
A)30.08 sq.units
B)45.11 sq.units
C)33.84 sq.units
D)41.35 sq.units
E)37.59 sq.units

Accepted Answer

To find the area of a triangle given two sides and the included angle, you can use the formula: Area = 1/2 * a * b * sin(C), where a and b are the sides, and C is the included angle. First, convert the angle C to decimal form: 55°43' = 55.717°. Then, use the formula with a = 6.5, b = 14, and C = 55.717° (in radians for the sine function). Area = 1/2 * 6.5 * 14 * sin(55.717° in radians) ≈ 37.59 square units.

Question 8

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. C = 73 ${}^\circ$ ,a = 12.5 and b = 11.5
?
A)68.96 sq.units
B)62.06 sq.units
C)55.17 sq.units
D)82.75 sq.units
E)75.85 sq.units

Accepted Answer

The area of a triangle can be calculated using the formula $ \frac{1}{2}ab\sin(C) $, where $a$ and $b$ are the lengths of two sides, and $C$ is the included angle. Substituting the given values, $a = 12.5$, $b = 11.5$, and $C = 73^\circ$, we get the area as approximately 68.96 square units.

Question 9

Find the area of the triangle having the indicated angle and sides.? $C = 102 ^ { \circ } , a = 6 , b = 2$ ? (Round your answer to one decimal place. )
?
A)?4.9
B)5.9
C)7.9
D)6.9
E)11.7

Accepted Answer

The area of a triangle given two sides and the included angle can be found using the formula: $\frac{1}{2}ab\sin(C)$, where $a$ and $b$ are the sides, and $C$ is the included angle. Plugging in the given values: $\frac{1}{2} \times 6 \times 2 \times \sin(102^\circ)$, and calculating this gives an area of approximately 5.9 square units.

Question 10

Find the area of the triangle having the indicated angle and sides. ​
A = 10°19',b = 4.6,с = 24
​
(Round your answer to one decimal place. )
​
A)9.9
B)8.9
C)10.9
D)11.9
E)19.8

Accepted Answer

The area of a triangle can be calculated using the formula: Area = 0.5 * b * c * sin(A). Converting 10°19' to decimal degrees gives approximately 10.317°. Using the formula, Area = 0.5 * 4.6 * 24 * sin(10.317°) ≈ 9.9.

Question 11

Because of prevailing winds,a tree grew so that it was leaning 6° from the vertical.At a point 43 meters from the tree,the angle of elevation to the top of the tree is 30° (see figure).Find the height a of the tree.​  ​c = 43 m B = 96°
​
(Round your answer to two decimal places. )
​
A)24.58 m
B)26.58 m
C)25.58 m
D)28.58 m
E)27.58 m

Accepted Answer

Let's call the height of the tree "a". Since the tree is leaning 6° from the vertical and we know the distance from the tree to point C (43 meters), we can use the tangent function to find the horizontal distance from the tree to point C.

tan(6°) = x/43

x = 43tan(6°) ≈ 4.55

So, the horizontal distance from the tree to point C is approximately 4.55 meters.

Now we can use the tangent function again to find the height of the tree.

tan(30°) = a/(43 + 4.55)

a ≈ (43 + 4.55)tan(30°) ≈ 26.58 meters

Therefore, the height of the tree is approximately 26.58 meters, which corresponds to choice B.

Question 12

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. B = 84°38',a = 7,and c = 8
A)33.45 sq.units
B)27.88 sq.units
C)25.09 sq.units
D)30.66 sq.units
E)22.30 sq.units

Accepted Answer

The area of a triangle can be found using the formula $Area = \frac{1}{2}ac\sin(B)$, where $a$ and $c$ are sides of the triangle, and $B$ is the angle between those sides. Given $a = 7$, $c = 8$, and $B = 84°38'$ (which is approximately $84.63°$ in decimal), we can calculate the area as follows:$$Area = \frac{1}{2} \times 7 \times 8 \times \sin(84.63°)$$Using a calculator to find $\sin(84.63°)$ and performing the multiplication:$$Area ≈ \frac{1}{2} \times 7 \times 8 \times 0.9952 ≈ 27.88$$Therefore, the area of the triangle is approximately $27.88$ square units.

Question 13

Find values for b such that the triangle has no solutions. ?
A = 64°,a = 340.2
?
A) $b \leq \frac { 340.2 } { \sin 64 ^ { \circ } }$
B) $b = \frac { 340.2 } { \sin 64 ^ { \circ } }$
C) $b \geq \frac { 340.2 } { \sin 64 ^ { \circ } }$
D) $b < \frac { 340.2 } { \sin 64 ^ { \circ } }$
E) $b > \frac { 340.2 } { \sin 64 ^ { \circ } }$

Accepted Answer

The answer of Find values for b such that the...

Question 14

Find the area of the triangle having the indicated angle and sides. ​
B = 123°,a = 64,c = 28
​
(Round your answer to one decimal place. )
​
A)750.4
B)753.4
C)1,502.9
D)752.4
E)751.4

Accepted Answer

To find the area of the triangle, we can use the formula: Area = 0.5 * a * c * sin(B). Substituting the given values: Area = 0.5 * 64 * 28 * sin(123°). Calculating this gives approximately 751.4.

Question 15

Select a value for b such that a triangle with A = 59° and a = 14 has only one solution. ​
A)b = 16
B)b = 17
C)b = 11
D)b = 15
E)b = 24

Accepted Answer

To have only one solution for the triangle, we need to make sure that the sine of the angle opposite to the given side is less than 1. This is because the sine function has a maximum value of 1. Using the sine law, we have:

b/sin(B) = a/sin(A)

Substituting the given values, we get:

b/sin(B) = 14/sin(59°)

To make sure there is only one solution, we need to make sure that sin(B) is less than or equal to 1. This means we need to choose a value of b such that:

b ≤ 14/sin(59°) ≈ 16.57

The only choice that satisfies this condition is C: b = 11.

Question 16

Find values for b such that the triangle has two solutions. ? A = 66°,a = 307.6 ? A) $307.6 < b \leq \frac { 307.6 } { \sin 66 ^ { \circ } }$ B) $307.6 \geq b \geq \frac { 307.6 } { \sin 66 ^ { \circ } }$ C) $307.6 > b > \frac { 307.6 } { \sin 66 ^ { \circ } }$ D) $307.6 < b < \frac { 307.6 } { \sin 66 ^ { \circ } }$ E) $307.6 \leq b \leq \frac { 307.6 } { \sin 66 ^ { \circ } }$

Accepted Answer

For a triangle to have two solutions in the ambiguous case (SSA), the side b must be greater than a but less than the length calculated by dividing a by the sine of angle A.

Question 17

After a severe storm,three sisters,April,May,and June,stood on their front porch and noticed that the tree in their front yard was leaning 3° from vertical toward the house.From the porch,which is 96 feet away from the base of the tree,they noticed that the angle of elevation to the top of the tree was 31°.Approximate the height of the tree.Round answer to two decimal places. ​
A)61.76 feet
B)69.64 feet
C)56 feet
D)59.13 feet
E)59.64 feet

Accepted Answer

The answer of After a severe storm,three sisters,April,May,and June,stood on...

Question 18

Find the area of the triangle having the indicated angle and sides. ​
B = 71°20',a = 62,c = 24
​
(Round your answer to one decimal place. )
​
A)705.9
B)​703.9
C)​706.9
D)​704.9
E)​1,409.7

Accepted Answer

The answer of Find the area of the triangle having...

Question 19

Find the area of the triangle having the indicated angle and sides. ​
С = 84°34',a = 16,b = 26
​
(Round your answer to one decimal place. )
​
A)209.1
B)​207.1
C)​208.1
D)​206.1
E)​414.1

Accepted Answer

To find the area of the triangle, we use the formula: 
Area = (1/2) * a * b * sin(C) 
where C is the given angle, and a and b are the given sides. However, before using this formula, we need to convert the given angle in degrees and minutes into decimal degrees. 
84°34' = 84 + 34/60 = 84.567 degrees (rounded to 3 decimal places)

Now we can substitute the given values into the formula and calculate the area: 
Area = (1/2) * 16 * 26 * sin(84.567) 
Area ≈ 207.1 (rounded to 1 decimal place)

Therefore, the best choice is B.

Question 20

Select a value for b such that a triangle with A = 33° and a = 4 has no solution. 
A)1.84
B)3.67
C)7.24
D)14.68
E)1.47

Accepted Answer

To have no solution for a triangle with A = 33° and a = 4, we need b to be greater than or equal to 8. This can be found using the sine rule: b/sin(B) = a/sin(A), where B is the angle opposite side b. Solving for sin(B), we get sin(B) = (b*sin(A))/a. Since sin(B) must be less than or equal to 1, we can solve for the maximum value of b: b = (a*sin(B))/sin(A). Plugging in the given values, we get b = (4*sin(B))/sin(33°). By testing the answer choices, we find that b = 14.68 is the smallest value that makes b greater than or equal to 8, so there will be no solution for a triangle with A = 33° and a = 4 if b = 14.68.

Quiz 36: Law of Sines

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. A = 129°,b = 8,and c = 11

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. A = 118 ${}^\circ$ ,b = 9 and c = 8. ?

A straight road makes an angle,A,of 10° with the horizontal.When the angle of elevation,B,of the sun is 56°,a vertical pole beside the road casts a shadow 8 feet long parallel to the road,see figure.Approximate the length of the pole.Round answer to two decimal places.

A park ranger at point A observes a fire in the direction N25°26'E.Another ranger at point B,4 miles due east of A,sites the same fire at N58°34'W.Determine the distance from point B to the fire.Round answer to two decimal places.

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. C = 55°43',a = 6.5,and b = 14

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. C = 73 ${}^\circ$ ,a = 12.5 and b = 11.5 ?

Find the area of the triangle having the indicated angle and sides.? $C = 102 ^ { \circ } , a = 6 , b = 2$ ? (Round your answer to one decimal place. ) ?

Find the area of the triangle having the indicated angle and sides. A = 10°19',b = 4.6,с = 24 (Round your answer to one decimal place. )

Because of prevailing winds,a tree grew so that it was leaning 6° from the vertical.At a point 43 meters from the tree,the angle of elevation to the top of the tree is 30° (see figure) .Find the height a of the tree. c = 43 m B = 96° (Round your answer to two decimal places. )

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. B = 84°38',a = 7,and c = 8

Find values for b such that the triangle has no solutions. ? A = 64°,a = 340.2 ?

Find the area of the triangle having the indicated angle and sides. B = 123°,a = 64,c = 28 (Round your answer to one decimal place. )

Select a value for b such that a triangle with A = 59° and a = 14 has only one solution.

Find values for b such that the triangle has two solutions. ? A = 66°,a = 307.6 ?

Find the area of the triangle having the indicated angle and sides. B = 71°20',a = 62,c = 24 (Round your answer to one decimal place. )

Find the area of the triangle having the indicated angle and sides. С = 84°34',a = 16,b = 26 (Round your answer to one decimal place. )

Select a value for b such that a triangle with A = 33° and a = 4 has no solution.

Quiz 36: Law of Sines

In the figure,? and ? are positive angles.? ? a = 9,b = 18 Write ? as a function of ?. ?

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. A = 129°,b = 8,and c = 11

In the figure,? and ? are positive angles.? ? a = 3,b = 12 Use ? = arcsin(0.25sin?) to write c as a function of ?. ?

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. A = 118 ∘{}^\circ∘ ,b = 9 and c = 8. ?

A straight road makes an angle,A,of 10° with the horizontal.When the angle of elevation,B,of the sun is 56°,a vertical pole beside the road casts a shadow 8 feet long parallel to the road,see figure.Approximate the length of the pole.Round answer to two decimal places.

A park ranger at point A observes a fire in the direction N25°26'E.Another ranger at point B,4 miles due east of A,sites the same fire at N58°34'W.Determine the distance from point B to the fire.Round answer to two decimal places.

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. C = 55°43',a = 6.5,and b = 14

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. C = 73 ∘{}^\circ∘ ,a = 12.5 and b = 11.5 ?

Find the area of the triangle having the indicated angle and sides.? C=102∘,a=6,b=2C = 102 ^ { \circ } , a = 6 , b = 2C=102∘,a=6,b=2 ? (Round your answer to one decimal place. ) ?

Find the area of the triangle having the indicated angle and sides. ​ A = 10°19',b = 4.6,с = 24 ​ (Round your answer to one decimal place. ) ​

Because of prevailing winds,a tree grew so that it was leaning 6° from the vertical.At a point 43 meters from the tree,the angle of elevation to the top of the tree is 30° (see figure) .Find the height a of the tree.​ ​c = 43 m B = 96° ​ (Round your answer to two decimal places. ) ​